Beyond the pooled mean: distributional structure in the attentional-focus meta-analysis

Meta-analyses of attentional focus in motor learning and performance have focused almost exclusively on pooled mean effect sizes, leaving potential distributional features under-examined. This study evaluates whether distributional tools applied to study-level effect-size posterior mixtures can capture evidence structures-such as directionally contrary evidence and shape consistency-that conventional mean-focused meta-analytic models obscure.

We reanalyzed 200 effect sizes from the published attentional-focus literature using three distributional frameworks applied to study-level effect-size posterior mixtures rather than participant-level raw distributions: Wasserstein-1 (W ₁) distances, a distributional heterogeneity measure H _w , and distributionally robust optimization (DRO). Against a zero-effect benchmark, W ₁ was decomposed into the pooled mean and a non-negative residual equal to twice the expected negative part of the mixture, quantifying the proportion of evidence opposing the recommended instructional focus. H _w was compared against the location-only prediction 1-I ²/100, and DRO stability margins (expressed as signal-to-noise ratios) were benchmarked against conventional trim-and-fill publication-bias corrections.

The directional residual derived from accounted for 36.4% of the transport distance for transfer outcomes, 20.5% for retention, and 13.5% for immediate performance; its bootstrap confidence intervals excluded zero across all evaluated outcomes, demonstrating that no sub-literature in this sample is fully summarized by its mean alone. The distributional heterogeneity measure exceeded location-only predictions across all five outcomes, most prominently for the distance effect (0.66 vs. 0.22), indicating greater agreement in distributional form than traditional variance-based indices suggest. Conversely, the outcome with the highest variance heterogeneity (I ² = 87.8% for EMG efficiency) exhibited the smallest directional residual (8.5%), as its contrary evidence lay close to the null point. DRO stability margins consistently exceeded trim-and-fill corrections across all primary outcomes.

Distributional metrics complement conventional pooled-mean comparisons by decoupling directional complexity from variance-based heterogeneity and providing robust stability bounds against data perturbation. However, because these analyses operate on study-level posterior mixtures, claims regarding individual participant-level tail behavior or extreme outcomes remain constrained and require access to raw participant-level data.